Skip to main content
SpringerLink
Log in

Surface Stress in an Elastic Plane with a Nearly Circular Hole

  • Chapter
  • First Online:
Book cover Surface Effects in Solid Mechanics

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 30))

Abstract

A boundary value problem on a nanometer hole in an elastic plane under arbitrary remote loading is solved. It is assumed that complementary surface stress is acting at the boundary of the hole. The outer surface of the hole is supposed to be conformally mapped on the outer surface of the circle by means of a power function. The Gurtin–Murdoch surface elasticity model is applied to take into account the surface stress effect. Based on the Goursat–Kolosov complex potentials and Muskhelishvili’s technique, the solution of the problem is reduced to a singular integro-differential equation in an unknown surface stress. For a nearly circular hole, the boundary perturbation method is used that leads to successive solutions of hypersingular integral equations. Numerical results based on the first-order approximate solution are specified for an elliptical nearly circular hole.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Altenbach, H., Eremeyev, V.A., Morozov, N.F.: Linear theory of shells taking into account surface stresses. Doklady Phys. 54, 531–535 (2009)

    Article  CAS  Google Scholar 

  2. Altenbach, H., Eremeev, V.A., Morozov, N.F.: On equations of the linear theory of shells with surface stresses taken into account. Mech. Solids 45, 331–342 (2010)

    Article  Google Scholar 

  3. Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49, 1294–1301 (2011)

    Article  CAS  Google Scholar 

  4. Duan, H.L., Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. Adv. Appl. Mech. 42, 1–68 (2009)

    Article  Google Scholar 

  5. Eremeyev, V.A., Morozov, N.F.: The effective stiffness of a nanoporous rod. Doklady Phys. 55, 279–282 (2010)

    Article  CAS  Google Scholar 

  6. Goldstein, R.V., Gorodtsov, V.A., Ustinov, K.V.: Effect of residual stress and surface elasticity on deformation of nanometer spherical inclusions in an elastic matrix. Phys. Mesomech. 13, 318–328 (2010)

    Article  Google Scholar 

  7. Grekov, M.A., Kostyrko, C.A.: Instability of a flat surface of a film coating due to surface diffusion. Vestnik St. Petersburg University, Ser. 10(1), 46–54 (2007) (Russian).

    Google Scholar 

  8. Grekov, M.A., Makarov, S.N.: Stress concentration near a slightly curved part of an elastic body surface. Mech. Solids 39(6), 40–46 (2004)

    Google Scholar 

  9. Grekov, M.A., Morozov, N.F.: Solution of the Kirsch problem in view of surface stresses. Mem. Differ. Equ. Math. Phys. 52, 123–129 (2011)

    Google Scholar 

  10. Grekov, M.A., Morozov, N.F.: Surface effects and problems of nanomechanics. J. Ningbo Univ. 25(1), 60–63 (2012)

    Google Scholar 

  11. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Rational Mech. Anal. 57, 146–147 (1975)

    Article  Google Scholar 

  12. Kaya, A.C., Erdogan, F.: On the solution of integral equations with strongly singular kernals. Quart. Appl. Math. 45(1), 105–122 (1987)

    Google Scholar 

  13. Linkov, A.M.: Boundary Integral Equations in Elastisity Theory. Kluwer Academic Publishers, Dordrecht (2002)

    Google Scholar 

  14. Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000)

    Article  CAS  Google Scholar 

  15. Murdoch, A.I.: A thermodynamic theory of elastic material interface. Q. J. Mech. Appl. Math. 29, 245–275 (1976)

    Article  Google Scholar 

  16. Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen (1963)

    Google Scholar 

  17. Povstenko, Y. Z.: Theoretical investigation of phenomena caused by heterogeneous surface tension in solids. J. Mech. Phys. Solids 41, 1499–1514 (1993)

    Google Scholar 

  18. Shenoy, V.B.: Atomic calculations of elastic properties of metalic fcc crystal surfaces. Phys. Rev. B 71, 94–104 (2005)

    Google Scholar 

  19. Tian, L., Rajapakse, R.K.N.D.: Analytical solution for size-dependent elastic field of a nanoscale circular inhomogeneity. Trans. ASME J. Appl. Mech. 74, 568–574 (2007)

    Article  Google Scholar 

  20. Tian, L., Rajapakse, R.K.N.D.: Elastic field of an isotropic matrix with nanoscale elliptical inhomogeneity. Int. J. Solids Struct. 44, 7988–8005 (2007)

    Article  CAS  Google Scholar 

  21. Vikulina, Y., Grekov, M.A., Kostyrko, S.A.: Model of film coating with weakly curved surface. Mech. Solids. 45(6), 778–788 (2010)

    Article  Google Scholar 

  22. Wang, J., Huang, Z., Duan, H., et al.: Surface stress effect in mechanics of nanostructured materials. Acta Mechanica Solida Sinica 24, 52–82 (2011)

    CAS  Google Scholar 

Download references

Acknowledgments

The work was supported by the Russian Foundation for Basic Research (grant 11-01-00230) and St.-Petersburg State University (project 9.37.129.2011).

Author information

Authors and Affiliations

  1. Saint-Petersburg State University, Universitetski pr. 35, 198504 , St.-Petersburg, Russia

    Mikhail A. Grekov

  2. Saint-Petersburg State University, Universitetski pr. 28, 198504 , St.-Petersburg, Russia

    Anna A. Yazovskaya

Authors
  1. Mikhail A. Grekov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna A. Yazovskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikhail A. Grekov .

Editor information

Editors and Affiliations

  1. , Lehrstuhl für Technische Mechanik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, Magdeburg, 39106, Germany

    Holm Altenbach

  2. St. Petersburg State University, Bibliotechnaya sq. 2, St. Petersburg, 198904, Russia

    Nikita F Morozov

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Grekov, M.A., Yazovskaya, A.A. (2013). Surface Stress in an Elastic Plane with a Nearly Circular Hole. In: Altenbach, H., Morozov, N. (eds) Surface Effects in Solid Mechanics. Advanced Structured Materials, vol 30. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35783-1_7

Download citation

Publish with us

Policies and ethics

Search

Navigation